newton's laws of motion - wordpress.com · newton’s laws of motion principia 1. every body...
TRANSCRIPT
Miss Millie Millie
Newton’s Laws of Motion Principia
1. Everybodycontinuesinitsstateofrest,oruniformmotioninastraightline,unlesscompelledtochangethatstatebyforcesacteduponit.
2. Thechangeofmotion(momentum)ofanobjectisproportionaltotheforceimpresseduponit,andismadeinthedirectioninwhichthefibreisimpressed.
3. Toeveryactionthereisalwaysanopposedorequalreaction.
Galilean transformations
𝑥 = 𝑥! + 𝑣𝑡 𝑦 = 𝑦′𝑧 = 𝑧′
!"
!"= !"!
!"+ 𝑣 𝑎 = 𝑎′
Whatifthesecondframeacceleratesw.r.tthefirst?
𝑎! = 𝑎 −𝑑𝑢𝑑𝑡
𝑧’
𝑦’
𝑥’
𝑧
𝑦
𝑥
𝑃
�̃�
�̃�′
𝑢!𝑡
𝑢
conservative forces ~Areindependentofpath~
𝑓! = −∇ 𝑈
! 𝑓!!!
!!∙ 𝑑�̃� = −! ∇ 𝑈
!
! 𝑑�̃� = [𝑈]!
!
= 𝑈(𝑄) − 𝑈(𝑃)
∴ ∆𝐾 = −∆𝑈
𝐸 = 𝐾 + 𝑈 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
E
U
KU x
y
Dot Product
𝑎! ∙ 𝑏! = |𝑎!|!𝑏!! cos 𝜃
𝑖𝑓 𝑎! ∙ 𝑏! = 0 𝑎! ⊥ 𝑏!
𝑎! = 0 & 𝑏! = 0
𝑎! ∙ 𝑎! = 𝑎!!
𝜃
𝑎!
𝑏!
3rd Law
𝑓!,! = −𝑓!,!~aparticlecannotexertaforceonitself~
𝑓! = −𝑓! 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑓! = 0
𝑚! 𝑚!"
Conservation of linear momentum Inasystemofmutuallyinteractingobjects,fortheithobject:
𝑚!𝑑!𝑟!𝑑𝑡! = ! 𝑓!" + 𝑓!"# =→ 𝑎𝑙𝑙 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒𝑠 𝑐𝑎𝑛𝑐𝑒𝑙 𝑎𝑛𝑑 𝑓!"# = 0
AngularMomentum
𝑃 = !𝑚!𝑑𝑟!𝑑𝑡 → 𝑡𝑜𝑡𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑙! = 𝑚!𝑟!𝑑𝑟!𝑑𝑡
∴ 𝐿 = ! 𝑙! → 𝑇𝑜𝑡𝑎𝑙 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 0 (𝑢𝑛𝑙𝑒𝑠𝑠 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑓!"#)
2nd Law 𝑑𝑝𝑑𝑡 = 𝑓 = 𝑚
𝑑𝑣𝑑𝑡 = 𝑚
𝑑!𝑥𝑑𝑡
𝑣! ∙ 𝑓! = 𝑣! ∙ !𝑚𝑑𝑣𝑑𝑡! = 𝑚𝑣!
𝑑𝑣𝑑𝑡
=𝑚2
! 2𝑣!𝑑𝑣𝑑𝑡! =
𝑚2!𝑑𝑣𝑑𝑡𝑣! +
𝑑𝑣𝑑𝑡𝑣!! =
𝑚2𝑑𝑑𝑡
(𝑣! ∙ 𝑣!) =𝑑𝑑𝑡𝑚𝑣!
2=𝑑𝐾𝑑𝑡
∆𝐾 = ! 𝑓 ! ∙ 𝑣!!!
!! 𝑑𝑡 = ! 𝑓!
!
!∙ 𝑑�̃� =𝑊
Non-isolated systems Inasystemofmutuallyinteractingobjects,withanexternalforce,fortheithobject:
𝑚!𝑑!𝑟!𝑑𝑡! = ! 𝑓!" + 𝑓!"#
𝑑𝑃𝑑𝑡 = 𝑓!"# = ! 𝑓!"#(!)
Totalsystemmomentumdependson𝑓!"# Torque
𝑑𝐿𝑑𝑡 = 𝜏 = ! 𝑟! × 𝑓!"#(!)